Czechoslovak Mathematical Journal, Vol. 62, No. 4, pp. 869-878, 2012

The $M_\alpha$ and $C$-integrals

Jae Myung Park, Hyung Won Ryu, Hoe Kyoung Lee, Deuk Ho Lee

Jae Myung Park, Hyung Won Ryu, Hoe Kyoung Lee, Department of Mathematics, Chungnam National University, Daejeon 305-764, South Korea, e-mail: parkjm@cnu.ac.kr; Deuk Ho Lee, Department of Mathematics Education, Kongju National University, Kongju 314-701, South Korea, e-mail: dhlee2@kongju.ac.kr

Abstract: In this paper, we define the $M_\alpha$-integral of real-valued functions defined on an interval $[a,b]$ and investigate important properties of the $M_{\alpha}$-integral. In particular, we show that a function $f [a,b]\rightarrow R$ is $M_{\alpha}$-integrable on $[a,b]$ if and only if there exists an $ACG_{\alpha}$ function $F$ such that $F'=f$ almost everywhere on $[a,b]$. It can be seen easily that every McShane integrable function on $[a,b]$ is $M_{\alpha}$-integrable and every $M_{\alpha}$-integrable function on $[a,b]$ is Henstock integrable. In addition, we show that the $M_{\alpha}$-integral is equivalent to the $C$-integral.

Keywords: $M_\alpha$-integral, $ACG_\alpha$ function

Classification (MSC 2010): 26A39


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